A logarithm is another way to write an exponential function. For example another was to write 10²=100 is log10100=2. Here is a diagram of what each term means:

    Logarithms were first used to simplify arithmatical processes. Now, logarithms are used in various applications of math. They are found everywhere; the shape a seaahell is a logarithmic spiral, the ricter scale of an earthquake is a logarithmic equation, the rate of depressiation, etc.

    The first logarithmic table to be published was introduced by John Napier, a Scottish mathmatician, in 1614. It was published again by a Swiss mathmatician, Justus Brygius, in 1620. The first common logarithmic table was compiled by the English man, Henry Briggs. A common logarithm is when the base of it is 10.  If a logarithm as no base, then the based is implied to be equal to ten. (i.e. log134=log10134) A natural logarithm is when the base of the the logarithm is equal to the none terminating number e. e is approximatly equal to 2.718281828. Although it seems it has a pattern, e is irrational. A logarithm with base e is written ln. (i.e. loge145=ln145)

    The log of a number doesn't always equal a happy little number. Although log100 is equal to 2, the log109 is roughly equal to 2.0374. Usually, a log is carried to four decimal places. The digits to the right of the decimal of a log is called the mantissa, while the digit(s) to the left of the decimal is called the characteristic. Logs can have the same mantissa but different chararteristic and visa versa (i.e. log109=2.0374 and log1090=3.0374)

There are 3 basic rules of logarithms which are:

Let Z be all positve reals except when Zis not equal to 1, let A equal all real numbers, let X and Y be positive real numbers.

Rule 1: Logz(xy) = logz(x) + logz(y)

Rule 2: Logz(x/y) = logz(x) - logz(y)

Rule 3: Logz(xª) = a logz(x)

Rule 1:     log(4x6) = log4 + log6

                1.3802  = .6021 + .7781

                1.3082  = 1.3082

Rule 2:     log(4/6) = log4 - log6

                -.1760  = .6021 - .7781

                -.1760  = -.1760

Rule 3.     log4^6  = 6 x log4

                3.6124 = 6 x .6021

                3.6124 = 3.6124

(values aren't exact)
 

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